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August  2020, 13(8): 2109-2120. doi: 10.3934/dcdss.2020180

Prey-predator model with nonlocal and global consumption in the prey dynamics

Malay Banerjee 1, , Nayana Mukherjee 1, and  Vitaly Volpert 2,3,4,,

1. 

Department of Mathematics & Statistics, Indian Institute of Technology Kanpur, Kanpur - 208016, India
2. 

Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622 Villeurbanne, France
3. 

INRIA, Université de Lyon, Université Lyon 1, Institut Camille Jordan, 43 Bd. du 11 Novembre 1918, 69200 Villeurbanne Cedex, France
4. 

Peoples Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

* Corresponding author: Vitaly Volpert

Received  January 2019 Published  August 2020 Early access  November 2019

Fund Project: The third author is supported by "RUDN University Program 5-100"

A prey-predator model with a nonlocal or global consumption of resources by prey is studied. Linear stability analysis about the homogeneous in space stationary solution is carried out to determine the conditions of the bifurcation of stationary and moving pulses in the case of global consumption. Their existence is confirmed in numerical simulations. Periodic travelling waves and multiple pulses are observed for the nonlocal consumption.

Keywords: Global consumption, pulse, traveling wave, speciation.
Mathematics Subject Classification: Primary: 35K57.
Citation: Malay Banerjee, Nayana Mukherjee, Vitaly Volpert. Prey-predator model with nonlocal and global consumption in the prey dynamics. Discrete & Continuous Dynamical Systems - S, 2020, 13 (8) : 2109-2120. doi: 10.3934/dcdss.2020180
References:
[1]

A. Apreutesei, A. Ducrot and V. Volpert, Competition of species with intra-specific competition, Math. Model. Nat. Phenom., 3 (2008), 1-27.  doi: 10.1051/mmnp:2008068.  Google Scholar

[2]

N. Apreutesei, A. Ducrot and V. Volpert, Travelling waves for integro-differential equations in population dynamics, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 541-561.  doi: 10.3934/dcdsb.2009.11.541.  Google Scholar

[3]

N. Apreutesei, N. Bessonov, V. Volpert and V. Vougalter, Sptaial structures and generalized travelling waves for an integro-differential equation, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 537-557.  doi: 10.3934/dcdsb.2010.13.537.  Google Scholar

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O. Aydogmus, Patterns and transitions to instability in an intraspecific competition model with nonlocal diffusion and interaction, Math. Model. Nat. Phenom., 10 (2015), 17-19.  doi: 10.1051/mmnp/201510603.  Google Scholar

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M. Banerjee and S. Banerjee, Turing instabilities and spatio-temporal chaos in ratio-dependent Holling-Tanner model, Math. Biosci., 236 (2012), 64-76.  doi: 10.1016/j.mbs.2011.12.005.  Google Scholar

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M. Banerjee, N. Mukherjee and V. Volpert, Prey-predator model with a nonlocal bistable dynamics of prey, Mathematics, 6 (2018), 1-13.  doi: 10.3390/math6030041.  Google Scholar

[7]

M. Banerjee and S. Petrovskii, Self-organized spatial patterns and chaos in a ratio-dependent predator-prey system, Theor. Ecol., 4 (2011), 37-53.   Google Scholar

[8]

M. Banerjee and V. Volpert, Prey-predator model with a nonlocal consumption of prey, Chaos, 26 (2016), 12pp. doi: 10.1063/1.4961248.  Google Scholar

[9]

M. Banerjee and V. Volpert, Spatio-temporal pattern formation in Rosenzweig-McArthur model: Effect of nonlocal interactions, Ecol. Complex., 30 (2017), 2-10.   Google Scholar

[10]

M. Banerjee, V. Vougalter and V. Volpert, Doubly nonlocal reaction–diffusion equations and the emergence of species, Appl. Math. Model., 42 (2017), 591-599.  doi: 10.1016/j.apm.2016.10.041.  Google Scholar

[11]

M. Baurmann, W. Ebenhoh and U. Feudel, Turing instabilities and pattern formation in a benthic nutrient-microorganism system, Math. Biosci. Eng., 1 (2004), 111-130.  doi: 10.3934/mbe.2004.1.111.  Google Scholar

[12]

M. Baurmann, T. Gross and U. Feudel, Instabilities in spatially extended predator-prey systems: Spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations, J. Theoret. Biol., 245 (2007), 220-229.  doi: 10.1016/j.jtbi.2006.09.036.  Google Scholar

[13]

A. Bayliss and V. A. Volpert, Patterns for competing populations with species specific nonlocal coupling, Math. Model. Nat. Phenom., 10 (2015), 30-47.  doi: 10.1051/mmnp/201510604.  Google Scholar

[14]

N. Bessonov, N. Reinberg and V. Volpert, Mathematics of Darwin's diagram, Math. Model. Nat. Phenom., 9 (2014), 5-25.  doi: 10.1051/mmnp/20149302.  Google Scholar

[15]

S. Fasani and S. Rinaldi, Factors promoting or inhibiting Turing instability in spatially extended prey-predator systems, Ecol. Model., 222 (2011), 3449-3452.  doi: 10.1016/j.ecolmodel.2011.07.002.  Google Scholar

[16]

T. Galochkina, M. Marion and V. Volpert, Initiation of reaction-diffusion waves of blood coagulation, Phys. D, 376-377 (2018), 160-170.  doi: 10.1016/j.physd.2017.11.006.  Google Scholar

[17]

G. F. Gause, The Struggle for Existence, Williams and Wilkins, Baltimore, 1934. doi: 10.5962/bhl.title.4489.  Google Scholar

[18]

S. Genieys, N. Bessonov and V. Volpert, Mathematical model of evolutionary branching, Math. Comput. Modelling, 49 (2009), 2109-2115.  doi: 10.1016/j.mcm.2008.07.018.  Google Scholar

[19]

S. Genieys, V. Volpert and P. Auger, Pattern and waves for a model in population dynamics with nonlocal consumption of resources, Math. Model. Nat. Phenom., 1 (2006), 65-82.  doi: 10.1051/mmnp:2006004.  Google Scholar

[20]

S. Genieys, V. Volpert and P. Auger, Adaptive dynamics: Modelling Darwin's divergence principle, Comp. Ren. Biol., 329 (2006), 876-879.  doi: 10.1016/j.crvi.2006.08.006.  Google Scholar

[21]

J. D. Murray, Mathematical Biology. Ⅱ: Spatial Models And Biomedical Applications, Interdisciplinary Applied Mathematics, 19, Springer-Verlag, New York, 2003. doi: 10.1007/b98869.  Google Scholar

[22]

S. Pal, S. Ghorai and M. Banerjee, Analysis of a prey-predator model with non-local interaction in the prey population, Bull. Math. Biol., 80 (2018), 906-925.  doi: 10.1007/s11538-018-0410-x.  Google Scholar

[23]

S. V. Petrovskii and H. Malchow, A minimal model of pattern formation in a prey-predator system, Math. Comput. Modelling, 29 (1999), 49-63.  doi: 10.1016/S0895-7177(99)00070-9.  Google Scholar

[24]

J. A. Sherratt, B. T. Eagan and M. A. Lewis, Oscillations and chaos behind predator-prey invasion: Mathematical artifact or ecological reality?, Phil. Trans. R. Soc. Lond. B, 352 (1997), 21-38.  doi: 10.1098/rstb.1997.0003.  Google Scholar

[25]

V. Volpert, Branching and aggregation in self-reproducing systems, in MMCS, Mathematical Modelling of Complex Systems, ESAIM Proc. Surveys, 47, EDP Sci., Les Ulis, 2014,116–129. doi: 10.1051/proc/201447007.  Google Scholar

[26]

V. Volpert, Elliptic Partial Differential Equations, Monographs in Mathematics, 104, Birkhäuser/Springer Basel AG, Basel, 2014. doi: 10.1007/978-3-0348-0813-2.  Google Scholar

[27]

V. Volpert, Pulses and waves for a bistable nonlocal reaction-diffusion equation, Appl. Math. Lett., 44 (2015), 21-25.  doi: 10.1016/j.aml.2014.12.011.  Google Scholar

show all references

References:
[1]

A. Apreutesei, A. Ducrot and V. Volpert, Competition of species with intra-specific competition, Math. Model. Nat. Phenom., 3 (2008), 1-27.  doi: 10.1051/mmnp:2008068.  Google Scholar

[2]

N. Apreutesei, A. Ducrot and V. Volpert, Travelling waves for integro-differential equations in population dynamics, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 541-561.  doi: 10.3934/dcdsb.2009.11.541.  Google Scholar

[3]

N. Apreutesei, N. Bessonov, V. Volpert and V. Vougalter, Sptaial structures and generalized travelling waves for an integro-differential equation, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 537-557.  doi: 10.3934/dcdsb.2010.13.537.  Google Scholar

[4]

O. Aydogmus, Patterns and transitions to instability in an intraspecific competition model with nonlocal diffusion and interaction, Math. Model. Nat. Phenom., 10 (2015), 17-19.  doi: 10.1051/mmnp/201510603.  Google Scholar

[5]

M. Banerjee and S. Banerjee, Turing instabilities and spatio-temporal chaos in ratio-dependent Holling-Tanner model, Math. Biosci., 236 (2012), 64-76.  doi: 10.1016/j.mbs.2011.12.005.  Google Scholar

[6]

M. Banerjee, N. Mukherjee and V. Volpert, Prey-predator model with a nonlocal bistable dynamics of prey, Mathematics, 6 (2018), 1-13.  doi: 10.3390/math6030041.  Google Scholar

[7]

M. Banerjee and S. Petrovskii, Self-organized spatial patterns and chaos in a ratio-dependent predator-prey system, Theor. Ecol., 4 (2011), 37-53.   Google Scholar

[8]

M. Banerjee and V. Volpert, Prey-predator model with a nonlocal consumption of prey, Chaos, 26 (2016), 12pp. doi: 10.1063/1.4961248.  Google Scholar

[9]

M. Banerjee and V. Volpert, Spatio-temporal pattern formation in Rosenzweig-McArthur model: Effect of nonlocal interactions, Ecol. Complex., 30 (2017), 2-10.   Google Scholar

[10]

M. Banerjee, V. Vougalter and V. Volpert, Doubly nonlocal reaction–diffusion equations and the emergence of species, Appl. Math. Model., 42 (2017), 591-599.  doi: 10.1016/j.apm.2016.10.041.  Google Scholar

[11]

M. Baurmann, W. Ebenhoh and U. Feudel, Turing instabilities and pattern formation in a benthic nutrient-microorganism system, Math. Biosci. Eng., 1 (2004), 111-130.  doi: 10.3934/mbe.2004.1.111.  Google Scholar

[12]

M. Baurmann, T. Gross and U. Feudel, Instabilities in spatially extended predator-prey systems: Spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations, J. Theoret. Biol., 245 (2007), 220-229.  doi: 10.1016/j.jtbi.2006.09.036.  Google Scholar

[13]

A. Bayliss and V. A. Volpert, Patterns for competing populations with species specific nonlocal coupling, Math. Model. Nat. Phenom., 10 (2015), 30-47.  doi: 10.1051/mmnp/201510604.  Google Scholar

[14]

N. Bessonov, N. Reinberg and V. Volpert, Mathematics of Darwin's diagram, Math. Model. Nat. Phenom., 9 (2014), 5-25.  doi: 10.1051/mmnp/20149302.  Google Scholar

[15]

S. Fasani and S. Rinaldi, Factors promoting or inhibiting Turing instability in spatially extended prey-predator systems, Ecol. Model., 222 (2011), 3449-3452.  doi: 10.1016/j.ecolmodel.2011.07.002.  Google Scholar

[16]

T. Galochkina, M. Marion and V. Volpert, Initiation of reaction-diffusion waves of blood coagulation, Phys. D, 376-377 (2018), 160-170.  doi: 10.1016/j.physd.2017.11.006.  Google Scholar

[17]

G. F. Gause, The Struggle for Existence, Williams and Wilkins, Baltimore, 1934. doi: 10.5962/bhl.title.4489.  Google Scholar

[18]

S. Genieys, N. Bessonov and V. Volpert, Mathematical model of evolutionary branching, Math. Comput. Modelling, 49 (2009), 2109-2115.  doi: 10.1016/j.mcm.2008.07.018.  Google Scholar

[19]

S. Genieys, V. Volpert and P. Auger, Pattern and waves for a model in population dynamics with nonlocal consumption of resources, Math. Model. Nat. Phenom., 1 (2006), 65-82.  doi: 10.1051/mmnp:2006004.  Google Scholar

[20]

S. Genieys, V. Volpert and P. Auger, Adaptive dynamics: Modelling Darwin's divergence principle, Comp. Ren. Biol., 329 (2006), 876-879.  doi: 10.1016/j.crvi.2006.08.006.  Google Scholar

[21]

J. D. Murray, Mathematical Biology. Ⅱ: Spatial Models And Biomedical Applications, Interdisciplinary Applied Mathematics, 19, Springer-Verlag, New York, 2003. doi: 10.1007/b98869.  Google Scholar

[22]

S. Pal, S. Ghorai and M. Banerjee, Analysis of a prey-predator model with non-local interaction in the prey population, Bull. Math. Biol., 80 (2018), 906-925.  doi: 10.1007/s11538-018-0410-x.  Google Scholar

[23]

S. V. Petrovskii and H. Malchow, A minimal model of pattern formation in a prey-predator system, Math. Comput. Modelling, 29 (1999), 49-63.  doi: 10.1016/S0895-7177(99)00070-9.  Google Scholar

[24]

J. A. Sherratt, B. T. Eagan and M. A. Lewis, Oscillations and chaos behind predator-prey invasion: Mathematical artifact or ecological reality?, Phil. Trans. R. Soc. Lond. B, 352 (1997), 21-38.  doi: 10.1098/rstb.1997.0003.  Google Scholar

[25]

V. Volpert, Branching and aggregation in self-reproducing systems, in MMCS, Mathematical Modelling of Complex Systems, ESAIM Proc. Surveys, 47, EDP Sci., Les Ulis, 2014,116–129. doi: 10.1051/proc/201447007.  Google Scholar

[26]

V. Volpert, Elliptic Partial Differential Equations, Monographs in Mathematics, 104, Birkhäuser/Springer Basel AG, Basel, 2014. doi: 10.1007/978-3-0348-0813-2.  Google Scholar

[27]

V. Volpert, Pulses and waves for a bistable nonlocal reaction-diffusion equation, Appl. Math. Lett., 44 (2015), 21-25.  doi: 10.1016/j.aml.2014.12.011.  Google Scholar

Figure 1.  Single pulse solution for prey and predator population for $ b = 10, d_1 = 0.1, d_2 = 0.1 $
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Figure 2.  Moving pulse for $ b = 10, d_1 = 0.3, d_2 = 0.1 $ (a) after time $ t = 500 $; (b) after time $ t = 600 $; (c) $ x $-$ t $ profile for $ L = 20 $
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Figure 3.  For two bifurcation diagrams, $ a = 1 $, $ \sigma_1 = 0.1 $, $ k_1 = k_2 = 0.35 $, $ \sigma_2 = 0.2 $ are fixed. (a) Bifurcation diagram in $ (L,d_1) $-plane for the values of parameters $ b = 5 $, $ \alpha = \beta = 0.35 $, $ d_2 = 0.1 $. (b) Bifurcation diagram in $ (d_1,b) $-plane for the values of other parameters $ \alpha = \beta = 0.363 $, $ d_2 = 1 $
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Figure 4.  Periodic travelling wave in the case of nonlocal consumption followed by a spatio-temporal structure. Left: a snapshot of solution with prey (blue) and predator (red) distributions. Right: level lines of the prey distribution $ u(x,t) $. The values of other parameters are $ a = 1,b = 1, \alpha = \beta = 0.363, \sigma_1 = 0.1, k_1 = k_2 = 0.35, \sigma_2 = 0.2, d_1 = d_2 = 1 $
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Figure 5.  Multiple moving pulses in the case of nonlocal consumption. Left: a snapshot of solution with prey (blue) and predator (red) distributions. Right: level lines of the prey distribution $ u(x,t) $. The values of other parameters are $ a = 1,b = 1, \alpha = \beta = 0.375, \sigma_1 = 0.1, k_1 = k_2 = 0.35, \sigma_2 = 0.2, d_1 = d_2 = 1 $
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Figure 6.  Different regimes observed in the case of nonlocal consumption presented on the $ (d_1,N) $ parameter plane for $ \alpha = 0.35 $ (left) and $ (N,\alpha) $ parameter plane for $ d_1 = 1 $ (right). The values of other parameters are $ a = 1,b = 1,\sigma_1 = 0.1, k_1 = k_2 = 0.35, \sigma_2 = 0.2, d_2 = 1 $
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